the Processes of Additive Combinatorics and Belief Revision
to Address Intractable Disagreements and Cognitive Dissonance
1. Additive Combinatorics
Group theory is the study of
groups or algebraic structures in abstract algebra derived from number
theory, geometry, and algebraic equations. Permutation groups
were developed from the search for general solutions to complex polynomial
equations. In group theory, a wreath product is a specialized
product of two groups based on a semidirect product.
A semidirect product is a specific
method of putting two subgroups together, with one being a normal subgroup.
The resultant semidirect product is a generalized product were a new
object is added to objects already known. A semidirect product
is a Cartesian product as a set with a specific multiplication operation.
The Cartesian product is a direct product of sets.
A permutation group is a group
whose elements are permutations of a given set and whose group operation
is the composition of permutations in that group. Permutation
group usually means a subgroup of a symmetric group. Applying
a permutation group to the elements being permutated (group action)
is made toward the study of combinatorics in mathematics and physics.
Additive combinatorics focuses
on the additive properties of the finite subsets of an additive group.
That group can be finite or infinite. An infinite group can be
reduced to a finite group using the Ruzsa projection (from works by
Imre Z. Ruzsa). For example, a set x
of infinite integers can imbed x in the finite cyclic group without
losing any combinatorial information if one is interested in the sums
of no more than two elements of x
at a time. Moreover, a finite set x can be modeled in that
set by an equal set in a finite group. That projection method
has several important uses in additive combinatorics and especially
in the structure of belief revision in paraconsistent logic.
2. Belief Revision
Two contradictory propositions,
in a real sense, cannot be true at the same time.the Law of Non-Contradiction.
But it is reasonable for finite beings to structure a group of inconsistent
and non-trivial subset theories in the process of reaching a threshold
of evidentiary proof for or against an overarching proposition, as seen
in additive combinatorics of finite and infinite groups. Logical
systems satisfying that requirement are called paraconsistent logics.
Paraconsistent logic attempts
to challenge the logical principle of, ex contradictione quodlibet
(ECQ), anything follows from contradictory premises and deals with contradictions
in a discriminating manner. The principle of ECQ explosion in
most logics can be expressed by the premise (X Λ ¬ X),
conjunctive elimination (X), weakening (X
V Y), conjunctive elimination (¬ X), disjunctive syllogism
(Y), and the conclusion (therefore Y). Paraconsistent
logic is logic whose consequence is not deemed to necessarily be explosive
and abandons at least one of the common logic processes. Many
logisticians choose to reject the disjunctive syllogism. The idea
being that, if not X, then X is excluded, so the only
way X or Y could be true would be if Y were true.
But if X and not X can both be true at the same time,
the reasoning fails. (The common symbols in logic for "statements"
are p, q, and r.)
In a true sense, both X
and not X can't both be true at the same time. However,
in reality observations may indicate that subsets of both group X
and group not X may be true. Thus, when the finite observations
are deemed true for a group they need to be aligned by moving the subset(s)
to the appropriate side of the X or not X matrix.
This will form a consistent logic that either X is true and not
X is false or X is false and not X is true. The steps
for addressing an unknown are (1) recognize the possibility of the unknown,
(2) discriminate its group by either set X is true or set not
X is true, (3) observe, (4) build set X and set not X,
(5) and conclude when the threshold has been perceived to have been
crossed that either X is true or not X is true.
A group could be an entity
defined by its subsets ranging from the characteristics of physics concepts
of blackholes to the theological attributes for the infinite God of
the first cause of all effects. For example, the group of the
existence of blackholes would have the sets of X (blackholes exist from
evidence of mathematical proofs by Hawking, etc.) and not X (blackholes
do not exist from evidence of violations of laws of quantum physics
by Laughlin, Chapline, and professors at Case Western Reserve University,
In Anton P. Chekhov's play
titled Ivanov, the character Anna Petrovna converts
to Russian Christian Orthodox from Judaism. Belief revision is
usually intended to reflect such models of updating rational beliefs
held by cognitive agents. (The preceding definition was excerpted
from an email exchange between Graham Priest, author of writings
on paraconsistent logic and to be professor at City University of New
York in 2009, and Dallas F. Bell Jr. in October, 2008.)
In the case of blackholes,
the cognitive agents are the aforementioned physicists, the updating
of information are the new discoveries that now seem to contradict the
initial beliefs for the existence of black holes (X) or nonexistence
of blackholes (not X). Another example could be the rational
writer of a science book likely believes that his text is true but also
knows that most complex works contain nontruth for a variety of reasons.
That writer would be rational to somewhat inconsistently believe that
his book is both true to a point and on some level not true at the same
There should not be an intractable
disagreement (among rational agents) as long as there is not just a
simple exclusion of one of the axioms. (The preceding opinion
was expressed by Vladimir Voevodsky, winner of the 2002 Fields Medal,
in an email exchange with Dallas F. Bell Jr. during October, 2008.)
Sadly, some atheists now agree that effects can occur without a cause
thereby attempting to illegitimately add to their set of not X
for the premise of the existence of God. Such seems to be the
true nature of intractable disagreements.
It has been argued that intractable
disagreements exist due to third party interjections that unnaturally
cause intractable discourse. On a superficial level that involves
opposing self-interests, such an explanation may have some merit.
We know that there are immutable truths in our value-laden world, e.g.
2 + 2 = 4. If someone elects not to accept the truth that 2 +
2 = 4, then they would have an intractable disagreement with those that
do believe the reality. By that simple math example, it is evident
that fault lies in the believer that 2 + 2 does not equal 4. The
failure is either a lack of information, a deliberate error or some
Solipsism is the philosophical
position that only one's own experience can be known. The notion
is that the self (mind/soul) is the only thing that we can know exists.
This pseudo intellectual theory is often used in academia and is easily
disproved. If we created all music then we should be able to play
all musical instruments, but we can't. If we created all poetry
we could write with the same complexity found in all poems, but we can't.
If we were all that exists, we would not need language or vocabulary,
as used in poems, to communicate with ourselves. Since we humans
are not all that exists, (moral) standards are not meaningless and it
is important to have goals to search for truth. That quest will
likely lead to disagreements.
It is epistemologically important
to examine the theological sets for the core of most intractable disagreement.
The group of the existence of the God of the first cause of all effects
could consist of the following two sets.
X (God exists)
--Physical Natural Laws (truth, information, gravity, etc.)
--Natural Laws of Freewill
(not murder, not lie, not steal, etc.)
--intellect, purpose, personality,
--soul, salvation, atonement,
justice, love, mercy
X (God does not exist)
--no eternity, no immutability
--no Physical Natural Laws
(no truth, no information, no gravity, etc.)
--no Natural Laws of Freewill
(no murder, no lying, no stealing, etc.)
--no intellect, no purpose,
no personality, no wisdom, no grace
--no soul, no salvation, no
atonement, no justice, no love, no mercy
Those infinite sets of subsets
presented in a finite range reach a threshold of belief from symbiotic
logic, observation and experience (including such writings as the Pharisee
of the Jews),
and revelation (Bible). Set X is rationally accepted as
true and not X can't be rationally accepted as true.
Human souls, proven by consciousness etc., can interact with God based
on the subsets or attributes of the living creator God. Yet there
are many people that have ears but do not hear and have eyes but do
not see the evidence (Matt. 13:13-23). They will have an intractable
disagreement with believers unless a belief revision is made.
Jesus knew the woman at the
well was seeking atoning salvation from the Messiah. When Jesus
presented her with the new information that He was the Messiah, she
and many others had a belief revision and received salvation. (John
4:4-42). On the other hand, Jesus did not engage in debates initiated
by lawyers and Pharisees for the benefit of their belief revision.
He defeated their logic for the belief revision of others (Luke 10:23-37).
Jesus replied to Satan, identified by Jesus as the father of the Pharisees,
in the wilderness by being factual and self-denying according to God's
words which glorify the Father God (Matt. 4:3-11). Paul's Sermon
at Mars Hill sought to address his listeners that were seeking new information
concerning the unknown God. He declared that this was the God
of the first cause of all effects (Acts 17:18-34).
It is known that people will
have the tendency to defend their belief status quo unless their needs
and aspirations are factored into the argument. A recent paper
suggests that people assign significant future value to victory at auctions
over bids of other people even at a loss but do not behave this way
when the opposition is a nonhuman computer.
Benefits and features favoring
belief revision should be made in debate. It is known that if
their belief status quo is attacked, they will retreat into an impregnable
defensive posture. This will make belief revision a less likely
outcome and make an intractable disagreement a more likely outcome,
even if the new information causes cognitive dissonance. It is
the experience of Phillip
law professor/author and seasoned debater, that logical argument does
not convince anyone to change a core position. The goal is to
persuade those open to persuasion. (Phillip Johnson's opinion
was expressed in an email exchange with Dallas F. Bell
Jr. in November, 2008.)
4. Cognitive Dissonance
Cognitive dissonance describes
a psychological state where beliefs are at odds. The stress of
this circumstance motivates people to resolve the inconsistent issue.
Social psychologists specialize in the application of cognitive dissonance
for individuals in society. (This sentiment was expressed by
Paul Bloom, coauthor of cognitive dissonance studies at Yale University,
in an email exchange with Dallas F. Bell Jr. during October, 2008.)
Children and monkeys have been
studied to find how they rationalize their choices to avoid cognitive
dissonance. In one study, monkeys were given a choice of either
of three M&M colored candies; one red, one green, and one blue.
If the monkey chose red over blue, the monkey was given another choice
between blue and green. Almost two-thirds of the time the blue
M&M that was rejected the first time was rejected again and the
green M&M was chosen. The belief was that this choice was
due to not wanting to think that a wrong decision had been made with
the first choice to reject the blue M&M.
The interpretation of the monkey
study with M&Ms has been clouded by the Monty Hall game. In
this game, one of three doors is chosen for a prize by the player.
One of the remaining two doors will be opened by the host to reveal
a lesser prize. A second choice is made by the player of whether
to keep the door chosen first or to switch it for the remaining door.
The odds for the best prize favor switching to the last door but most
people seem to stay with the original door chosen even thought they
have less odds of receiving the best prize.
Cognitive dissonance can be
avoided by new information, such as statistical evidence. A method
was developed to make better group decisions called Delphi. Three
to five heterogeneous experts in different fields make a decision for
a course of action. Unfortunately, congresses are groups of elected
officials with no expertise in the areas they are expected to make decisions
about and expectedly make numerous laws that harm societal efficiency.
The jury system is closer to reflecting a Delphi process. (The
information on the Delphi method was excerpted from an email exchange
with Murray Turoff, Delphi coauthor and U. S. defense analyst, and Dallas
F. Bell Jr. in October, 2008.)
The 1908 book, Orthodoxy, by Gilbert K. Chesterton links
Christianity to promotion of the governmental system of democracy.
Chesterton explains how a person can come to believe the Christian faith
and does not waste the effort of attempting to show why it is to be
believed. He believed it is not an arbitrary truth from outside
the boundaries of human experience. Instead, Christianity is the
answer to natural human needs.
Additive combinatorics can
be used as a tool for groups. Belief revision is a natural process
for acceptance of new information. When new information is rejected
intractable disagreements can result as seen in the fictional story
of Dostoevsky's The
(found in the work The Brothers Karamazov) where Jesus was arrested
and confronted by the anti-Christ priest who is a latter day Pharisee.
Richard Dawkins' algorithm in his book, The Blind Watchmaker,
makes the same 'Faustian' (from a work by Johann W. Goethe)
error as Wolfram's algorithm to disprove the existence of creative
intellect and purpose (God). Humorously, their algorithms required
each of their purposeful intellects to be created. They hear but
do not understand and they see but they do not perceive (Is. 6:9).
That circumstance causes an unnecessary state of cognitive dissonance.
King Solomon wrote, "that
which has been is now, and that which is to be has already been, and
God requires that which is past because He will judge the righteous
and the wicked" (Eccl. 3:15-17).
RIGHTS RESERVED © 2008 DALLAS F. BELL, JR.---------------